• Data description is the process of summarizing and displaying the key features of a dataset
• It helps in understanding the data's characteristics, patterns, and potential insights

Statistic vs. Parameter

• A statistic is a characteristic or measure obtained using data values from a sample
• A parameter is a characteristic or measure obtained using all the data values from a specific population

Rounding Rule

• Rounding should be done only after the final answer is calculated to avoid early rounding error
• Use parentheses or spreadsheets to maintain accuracy during calculations

Data Description: Measures

• Measures of central tendency describe the typical or average value in a dataset
• Measures of variation (dispersion) indicate the spread or variability of the data
• Measures of position describe the relative location of specific data values within the dataset

Measures of Central Tendency

• An average is a measure of central tendency that summarizes the typical value of a dataset
• Common measures of central tendency are Mean, Median, Mode, Midrange and Weighted Mean

Mean

• The mean is the sum of the values divided by the total number of values
• It is denoted as X for a sample mean
• It is denoted as μ for a population mean
• For sample mean: X = (X₁ + X₂ + X₃ + ... + Xn) / n = (ΣX) / n
• For population mean: μ = (X₁ + X₂ + X₃ + ... + XN) / N = (ΣX) / N

Example Mean of Days off

• Problem is to calulate the mean number of days off per year for a sample of individuals selected from nine different countries with reported days off being 20, 26, 40, 36, 23, 42, 35, 24, 30
• The solution: X = (20 + 26 + 40 + 36 + 23 + 42 + 35 + 24 + 30) / 9 = 276 / 9 = 30.7
• The mean number of days off is 30.7 years

Rounding Rule for the Mean

• The mean should be rounded to one more decimal place than in the raw data
• The havaria refers to the amount of money that each person was called upon to pay for lost or damaged goods on a ship

Mean for Grouped Data

• The mean for grouped data is calculated by multiplying the frequencies by their midpoints of the classes
• The formula: X = Σ(f · Xm) / n, where Xm is the midpoint and f is the frequency of each class

Example Mean for Grouped Data

• The frequency distribution of miles run per week can be used to calculate the mean
• Columns include Class boundaries | Frequency, f | Midpoint, Xm | f.Xm |
• The solution: X = Σ(f · Xm) / n = 490 / 20 = 24.5 miles

Geometric Mean

• Geometric mean is defined as the nth root of the product of n values and is useful for percentages, ratios, indexes, or growth rates
• Formula: n√(X₁ · X₂ · ... · Xn)
• As an example the geometric mean of 1, 3, 9 is: ³√1 · 3 · 9 = 3

Median

• The median is the midpoint of the data array, symbolized as MD
• With an odd number of values, the median is one of the data values
• With an even number of values, the median is the average of two data values
• The concept of the median was introduced as a statistical concept by Francis Galton around 1874

Median Examples

• With values of rooms in seven hotels in downtown Pittsburgh including 713, 300, 618, 595, 311, 401, and 292, the median is found to be 401 rooms
• Sort in ascending order : 292, 300, 311, 401, 596, 618, 713
• With data regarding the number of tornados follows, for an 8-year period : 684, 764, 656, 702, 856, 1133, 1132, 1303
• After sorting, the median is found to be 810.
  • 656, 684, 702, 764, 856, 1132, 1133, 1303
  • MD = (764 + 856) / 2 = 1620 / 2 = 810

Mode

• The mode is the value that occurs most often in a dataset and is considered the most typical case
• Data can have no mode, one mode (unimodal), two modes (bimodal), or many modes (multimodal)
• The mode was first used by Karl Pearson in 1894.

Mode Examples

• Data includes the signing bonuses of eight NFL players in millions of dollars : 18.0, 14.0, 34.5, 10.0, 11.3, 10.0, 12.4, 10.0
  • Sorting the data results in 10.0, 10.0, 10.0, 11.3, 12.4, 14.0, 18.0, 34.5 and the mode turns out to be 10 million
• Data includes number of coal employees per county for 10 counties including, 110, 731, 1031, 84, 20, 118, 1162, 1977, 103, 752 and no mode exists
• Data regarding licensed nuclear reactors including = 104, 104, 104, 104, 104, 107, 109, 109, 109, 110, 109, 111, 112, 111, 109 and finds results for bimodal since it contains the modes 104 and 109
• Frequency distribution of miles that 20 runners ran for the modal class are shown
  • The class boundaries columns are shown as 5.5 - 10.5 = 1 | 10.5 - 15.5 = 2 | 15.5 - 20.5 = 3 | 20.5 - 25.5 = 5 | 25.5 - 30.5 = 4 | 30.5 - 35.5 = 3 | 35.5-40.5 = 2. In this class the modal class is found to be 20.5 – 25.5 with a midpoint of 23 miles a week

Midrange

• The midrange is the average of the lowest and highest values in a dataset.
• MR = (Lowest + Highest) / 2

Midrange Example

• Data on water-line breaks in Brownsville, Minnesota reports the water breaks per month including 2, 3, 6, 8, 4, 1
• Problem shows that the midrange is 4.5.
  • MR = (1 + 8) / 2 = 9 / 2 = 4.5

Weighted Mean

• Weighted mean of a variable is calculated by multiplying each value by its corresponding weight, then dividing by the sum of the weights
• Formula: X= (w₁X₁ + w₂X₂ + ... + wnXn) / (w₁ + w₂ + ... + wn) = ΣwX / Σw

GPA Example

• Example shows how to caluclate Grade Point Average (GPA) in the following classes
  • English Composition w/ a grade of A and 3 credits
  • Introduction to Psychology w/ a grade of C and 3 credits
  • Biology w/ a grade of B and 4 credits
  • Physical Education w/ a grade of D and credits of 2.
• The GPA is calculated to be = 2.7

Mean Properties

• Uses all data values
• Varies less than the median or mode when samples are taken from the same population
• Used in computing statistics such as the variance
• Unique, usually not one of the data values
• Cannot be used with open-ended classes
• Affected by outliers

Median Properties

• Gives the midpoint of data
• Used to find out it data values fall inupper / lower distribution
• Can be uses for open-ended distribution
• Affected less than the mean by extremely low or high values

Mode Properties

• Used when the most typical case is desired
• Easiest average to compute
• Uses for nominal Data
• Not always unique or may not always exist.

Midrange Properties

• Easy to compute
• Gives the midpoint of data
• Affected by extremely low or high values

Skewness

• Skewness is a measure to determine the skew of a distribution that is also know as the Pearson coefficient. PC = (3 * (X – MD)) / s, where S = Standard Deviation
• Coefficient values range from -3 to +3.
• symmetric when coefficient value nears 0
• when values are positvely skewed = postive value.
• when values are negativley skewed is when equation results in a negative answer.

Measures of Variation

• Measures of variation (dispersion) describe the spread or variability of data
• Range shows the spread
• Variance and Standard Deviation is used

Range

• Range is is calculated by finding the difference between the highest and lowest values in a data by set, Formula = R = Highest – Lowest

Range Example

• Experiment was conducted to observe two brands of outdoor paint for fading by testing six cans over a 3 to 6 month average to a small population
• Brand A showed its values as 10, 60, 50, 30, 40, 20
• Brand B showed its values as 35, 45, 30, 35, 40, 25
• The solution provided the details
• Brand A | mean (μ = 35) and its range value shows 50
  • R =60-10 = 50
• Brand B | mean (μ = 35) and its range value is 20
  • Ri =45-25 = 20

Variance and Standard Deviation

• Variance is the average of the squares of the distance each value is from the mean
• Standard deviation (σ for population, s for sample) is the square root of the variance
• The standard deviation indicates how spread out data is

Variance / Standard Deviation - Uses

• Determining spread of the data
  • when variance of SD is is large- data is dispersed
• Determine the consistency of the vaiable
  • for example (nuts and bolts) manufactured diameter fittings
    • must be small, or parts will not fit together
• determine the number of data values that fall within a distribution
• Used in interential statistcs

Variance & Standard Deviation (Population)

• Population variance: σ² = Σ(X - μ)² / N
• Population standard deviation: σ = √[Σ(X - μ)² / N]
• the variance calculation, is the standard deviation using same raw data
• Finding square root will allow numbers to be in same units
• Note: always use its positive value to its finding the square root.
• Rounding Rule:
  • Standard deviation rounding guidele = similar as the mean- final result should be rounded to one more decimal place tht the orignal data.

Variance & Standard Deviation Example

• Set of data includes the brand values for Brand A with: 10, 60, 50, 30, 40, 20 with a total mean of 35
  • standard deviation = 17.1 - (rounded)

Variance & Standard Deviation

• To find the variance and distribution for samples from computational models, use the theoretical formula.
  • Saves time in hand caculation
  • does not use the mean
  • provides accuate number if mean has been rounded.

Computational Variance-Sample

• Find/ save Variance and deviation in samples.
  • Used for samples - and uses computatuional models
  • the sample varance =s²= nΣX²_(ΣX)² / n(n-1)
  • samples standard deviation s=√s²

Variance & Standard Deviation Example - European

• For a sample of six years find the volume and the standard deviation amount of the European value with the numbers 11.2
  • 11.9,
  • 12.0,
  • 12.8,
  • 13.4,
  • 14.3 = amount in millions of dollars
  • answer shows:
  • Value is 1.28
  • Samples = 1.13

Grouped Data Variance/ Standard Deviations

• the procedure is similar - that of for for the mean for groups
  • Data and its uses uses midpoint in each.
  • n= is the sum number as the freqeunrcies

Variance / Standard summary of Deviation

• Finding Summary for Measurement and what their definitions are in symbols.
  • for range = shows distance/symbol =R
    • highest vs the lowest
  • Variance= average Squares in each, Symbol= σ2, 52
  • distant each- measured in the mean.
  • Deviation - measured from Squares- Variance Root. Symbol
  • Shows σ, S- square root in vaiaance

Coefficient of Deviation

• coefficient calculation= standard. Deviation/ expressed as (precent)
  • CVAR S/ X *100 precent
  • S= Standard deviation
  • X= mean = the number
• how does it compare to two standards - what it Deviats to?
• exsample heights - for both men and women!

Statistics for Deviations in Samples

• Carl Pearson devised - variation and comparing two different deviations (men or women - heights)
  • the deviatiaion is based on the population and deviations standards for population and means.

CVAR EXAMPLE DEVIATION PROBLEMS to work out

• Number of sales over (3- month ) - sales of (cars,) and 87 -Deviations = ( 5) 5225- Mean and numbers, Deviations? -773 What variation do the numbers - look like?
  • The answer.
    • Cvar= 5/87*100 precent= to 5.7
    • • for these numbers= Sales
  • Cvar= 773/5225*-100 precent to 14. 8 - these result numbers are Commisions
  • to come too the solution/ or the variance factor- that commisions is higher than samples.

RAnge and its Rules. / 4 when distributions are UNIMODAL

• range can show approximate numbers for the standard Deviations
  • thumb rules will show approximate # and as deviation and what's symmetrical is the Formula =S ( precent) = the ranged #
• devide(d) by 4- numbers / symmetrical

Range - Rules and tips

• Use _ /+- the 2xs - approximate in the low end , -High end of set data,
• example
  • set example =10 = the numbers or sample-
  • range the numbers equal 12
  • S = aprox 12/ or equals to approx 3!
  • set for the end - LOW = Aprox 10- 2 {3) =4. end of thumb scale!
    • higher scale numbers = Aprox is / 16 approx.

Rules of Deviation, theorem use/ approximation theam to equal them out

• using Chebyshev theorems for approximations- in empral- is used for normal distributions

Rule by Chebyshev with Theorem

• theorem = data shape / applied
  • regardless used set- shape.
• Data with k standard- can use to set deviations to means(U). with at lest an 1 K 2 where k equals to a number (than) # than 1. , is not or cannot be used as a interger.

How do - numbers equal using =Chebyshev?

• of the standard devitations-

  • K
• Proportions is the MINIUM ! Proportations.
  • inside/ set standars or used for measurements.
    • ex.
      • 1 1/4 =
    • 3/4 or 2 = ( 1-3/4 = .34)
      • or can be set and equal to percentages.
  • MIN numbers or Minimum standard= percentages - with standard devitations

Chebyshev Theorem and equal / non useful factors of set # 's

• it's Non useful . =1

• Formula for equal measurements=

  • P( lx ( minus) M 1 = the KO = 1- K

          -2
          = for example,
    

Chebyshev’ Theorem - example numbers used too find equations.

• The Mean set house price = 50000 Deviations Standard = 10000 prices will se
• Standard Devitation in MIN numbers =75 or
  • will set in deviation 2. Deviations of number
    • 50000 -( ( minus) 2*(10000) = 30000
    • the 10000 ( 2 + set / times (2)) = =70000
    • MIN amount of values set homes =to sell over 30000 and to prices of- 70000 with No knowledge what deviations will do!

Theorums with Chebyshev

• Example of Theorem
• Using the Deviation Theorem in - Miles 0.25 ( average mile)s, Deviatains Standard 002
• how find that MIN Numbers for Equal numbers of ranges
  • 0.20 and 0.30?
• With Equation: - (0.30(0minus).2 )/devite .02 = to 2.5
• with the other Equation used/ - ( .02.5-.27)/.02 = equal to
  • • 2.5 =

  • K= 2.5. or 1 - 1/ K *2- 1-1/equal 2.5 - 1- 1 equal- .084

  • *At lest 85 precent will the data is/ will fall=numbers that equal to .